An incremental approach to the statement and solution of the problem of the nonlinear deformation of shells under loads that cause buckling and strong bending in the plastic range is developed. The relations between strains and displacements for great angles of rotation are used. A system of differential equations for the rates of the unknown functions is derived and represented in Cauchy operator form. To solve the boundary-value problem, the discrete-orthogonalization method is used assuming that the unknown functions and the loads are equivalent. The problem of the buckling and postbuckling behavior of a long D16T-alloy shell with a local initial deflection is solved as an example Keywords: nonlinear deformation of shell, buckling, postbuckling behavior, strong bending in the plastic range, wide angles of rotation, discrete-orthogonalization methodIntroduction. The stability analysis of shells consists of three stages corresponding to three characteristic states observed in a shell during quasistatic loading. The initial (subcritical) stage of deformation and a state near the critical or bifurcation loads at small strains, deflections, and angles of rotation can in most cases be described with high accuracy by the nonlinear equations of the Donnell-Mushtari-Vlasov or Flugge theory [1,8,10]. The final (postcritical) stage of deformation, during which the geometry changes substantially, can be adequately described only by theories that do not restrict the magnitudes of the deflections and angles of rotation. There are several such nonlinear theories of shells (see [11] for their advantages and disadvantages). However, there is still no a generally accepted, well-founded theory of shells that would be valid for both small and large displacements. In addition to kinematic refinements, the applicability range of the theory can be expanded by selecting a constitutive equation that fits better than Hooke's law. This is because many materials display plastic behavior when even relatively thin shells made of such materials are subject to strong bending.An asymptotic analysis of the stability of structures in plastic state was carried out in [13]. Since a bifurcation load in the plastic range does not lead to buckling because the load is increased in the sense of Shanley, a study was made of how this affects the postcritical behavior and sensitivity to imperfections of compressible rods and shallow shells. The effect of plasticity in the case of substantial changes in geometry was numerically analyzed in [6,12].Currently, applications that stimulate studies in this field have intensively been developed [9,14,17,18].Here we will develop an incremental approach to the formulation and solution of the problem of the nonlinear deformation of shells subject to compressive loads that cause buckling and strong bending in the plastic range. It is based on nonlinear relations between strains and displacements that are applicable at great angles of rotation. They will be derived from the Novozhilov equations of nonlinear elasticit...